\input epsf
\nopagenumbers
\magnification=\magstep1

\hskip 3.56in\vbox{
\def\tablerule{\noalign{\hrule depth0pt height.5pt width1.575in}}
\halign to \hsize{ \hfil $\ # $\strut\quad \vrule\thinspace
& \hfil $ #
$\quad \vrule\thinspace &
\hfil $ # $\quad \vrule\thinspace
& \hfil $ # $\cr
\ x \negthinspace\hfil &\  f(x)
\negthinspace\negthinspace\negthinspace\hfil  &\
f'(x)\negthinspace\negthinspace\negthinspace\negthinspace  \hfil &\
f''(x)\negthinspace\negthinspace\negthinspace\negthinspace  \hfil\cr
\tablerule
\tablerule
1 & 2 & 0 & 2 \cr
\tablerule
2 & 3 & 6 & 5 \cr
\tablerule
3 & 7 & 3 & -4\cr
\tablerule
4 & 2 & 5 & 7 \cr}}

\vskip -.87in
\vtop{\hsize=3.73in
\noindent {\bf Problem statement} Values of a twice differentiable
function, $f$, and its first and second derivatives are in the table
to the right. Use this information to answer the following questions
as well as you can.
}

\medskip

\noindent a) If $g(x)= \bigl( f(x) \bigr)^{\! 2}$, compute $g(2)$, $g'(2)$,
and $g''(2)$.

\medskip

\noindent b) If $h(x)= f(x^2)$, compute $h(2)$, $h'(2)$, and $h''(2)$.

\medskip

\noindent c) If $k(x)= f(f(x))$, compute $k(2)$, $k'(2)$, and $k''(2)$.

\medskip

\noindent d) If $h$ is a small number, write an approximation (the
linearization of $f$ at $3$) for $f(3+h)$.

\medskip

\noindent e) Is your answer in d) likely to be an underestimate of the
true value of $f(3+h)$ when $h$ is small, or an overestimate? Give a
reason for your answer.


\vfil\eject\end

